Whether in the real world or the anime one, killing is a heinous act. Stealing someone else's future is something that nobody should want to do. Therefore, many people, real or not, try to find less permanent solutions to their conflicts.
Unfortunately, depending on the anime, that’s not always an option. Plenty of characters are forced into situations where taking someone else’s life is the only path available. This becomes even worse when it’s the protagonist who’s forced to resort to murder.
Anime characters react to killing someone in different ways depending on the circumstances. If it’s someone like Trigun's Vash the Stampede who spends the entire series preaching pacifism before being forced to take a life, it can destroy them on the inside and make them question all their life choices. One death in 25 episodes doesn’t sound like much, but to the guy who swore never to kill, it means everything.
Some protagonists are eased into killing people. This happens to a lot of Gundam pilots in the Mobile Suit Gundam franchise; many characters are dragged into a war they never wanted to fight and forced to destroy enemy mobile suit pilots. It gets really bad for them once they learn more about the people they’re fighting against and start to think deeply about how many of them they’ve killed.
Then there are the “heroes” who get used to killing far too quickly. They may be horrified or even disgusted by it the first time, but once they’ve convinced themselves it’s a necessary evil for achieving their goals, they do it routinely. These characters, including Death Note's Light Yagami and Lelouch Vi Brittania from Code Geass, might understand that killing is wrong, but they'll still kill as many as they have to for their cause.
Some protagonists feature in universes where killing and violence are inescapable parts of their lives. These protagonists live in harsh, unforgiving worlds of survival-of-the-fittest and kill-or-be-killed. For example, Guts (Berserk) and Caiman (Dorohedoro) probably wouldn’t kill if they didn’t have to, but that’s just the way things are for them.
To clarify, this doesn't apply to killing monsters and other nonhuman entities. Some anime suggest the possibility for friendly relationships with such creatures, but for the most part, they are presented as cold, unfeeling killing machines whose sole purpose is destruction. The protagonists of anime with these types of enemies have no qualms about killing them, nor should they. If these count as kills, then main characters like Ichigo (Bleach), Tanjiro (Demon Slayer) and any exorcist manga protagonists are responsible for a whole lot of murder.
This mindset can even be applied to some humans. If there are people who act inhumanly or derive pleasure from the suffering of others, they can’t really be seen as people. With these kinds of “humans,” main characters like Kenshiro (Fist of the North Star) and Tatsumi (Akame ga Kill!) can get away with killing whoever they fight and still be seen as heroes, both in-universe and out.
What really makes the difference is killing people. To take somebody’s life is to take away absolutely everything from that person. They don't just lose their ability to hurt others but also their passions, hobbies, dreams and everything else that was good about them. It’s especially devastating to those who knew and cared for them, which is part of why they often seek revenge. These consequences of killing and the motives behind them are what separate characters like Vash from a mass murderer like Attack on Titan's Eren Yeager.
A lot of battle shonen protagonists circumvent this issue with raw power. By becoming strong enough to stop or beat their opponents without needing to kill them, they can continue to pursue their dreams without having to think too hard about the blood on their hands. That’s not to say that shonen protagonists like Goku (Dragon Ball) or Meliodas (The Seven Deadly Sins) haven’t resorted to murder before -- only that it's not nearly as "necessary" as some others might deem it. This approach sounds rather optimistic, but also comforting.
Again, death should be as far down a person's list of solutions as possible. Anime protagonists who end up killing have all sorts of reasons and circumstances that make it an unavoidable part of their journeys; these circumstances should not be perceived as the standard. People should always seek to resolve their conflicts in the least violent way possible, even if that sounds idealistic.
